# Properties

 Label 5776.2.a.bt Level $5776$ Weight $2$ Character orbit 5776.a Self dual yes Analytic conductor $46.122$ Analytic rank $0$ Dimension $4$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5776 = 2^{4} \cdot 19^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5776.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$46.1215922075$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(\zeta_{20})^+$$ Defining polynomial: $$x^{4} - 5x^{2} + 5$$ x^4 - 5*x^2 + 5 Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 722) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## $q$-expansion

Coefficients of the $$q$$-expansion are expressed in terms of a basis $$1,\beta_1,\beta_2,\beta_3$$ for the coefficient ring described below. We also show the integral $$q$$-expansion of the trace form.

 $$f(q)$$ $$=$$ $$q + ( - \beta_{3} - \beta_{2} - 1) q^{3} + (\beta_{3} - \beta_{2} - \beta_1 - 1) q^{5} + ( - \beta_{2} + \beta_1) q^{7} + (2 \beta_1 + 1) q^{9}+O(q^{10})$$ q + (-b3 - b2 - 1) * q^3 + (b3 - b2 - b1 - 1) * q^5 + (-b2 + b1) * q^7 + (2*b1 + 1) * q^9 $$q + ( - \beta_{3} - \beta_{2} - 1) q^{3} + (\beta_{3} - \beta_{2} - \beta_1 - 1) q^{5} + ( - \beta_{2} + \beta_1) q^{7} + (2 \beta_1 + 1) q^{9} + ( - 3 \beta_{2} - \beta_1 - 2) q^{11} + ( - \beta_{3} - \beta_{2} - \beta_1 + 4) q^{13} + (\beta_{3} + 4 \beta_{2} + \beta_1 + 1) q^{15} + ( - \beta_{2} + 2 \beta_1 + 1) q^{17} + ( - 2 \beta_{3} - 2 \beta_{2}) q^{21} + (3 \beta_{3} - 3 \beta_{2} + 2 \beta_1 + 1) q^{23} + (2 \beta_{3} - 3 \beta_{2}) q^{25} + ( - 2 \beta_{2} - 2 \beta_1) q^{27} + ( - \beta_{3} - 3 \beta_{2} + \beta_1 - 2) q^{29} + (\beta_{3} - \beta_{2} - 7) q^{31} + (4 \beta_{2} + 4 \beta_1 + 6) q^{33} + (\beta_{3} + \beta_{2} - 2 \beta_1 - 1) q^{35} + ( - \beta_{3} - 2 \beta_1 + 1) q^{37} + ( - 4 \beta_{3} - 3 \beta_{2} + 3 \beta_1) q^{39} + ( - 2 \beta_{3} + 2 \beta_{2} - \beta_1 - 2) q^{41} + (\beta_{3} + \beta_{2} - 2 \beta_1 + 3) q^{43} + ( - \beta_{3} + \beta_{2} - 3 \beta_1 - 5) q^{45} + ( - \beta_{3} + 4 \beta_{2} - \beta_1 + 5) q^{47} + ( - 2 \beta_{3} - 3) q^{49} + ( - 4 \beta_{3} - 5 \beta_{2} - \beta_1 - 2) q^{51} + (3 \beta_{3} - 2 \beta_{2} + 1) q^{53} + (5 \beta_{3} + \beta_{2} + 7) q^{55} + ( - \beta_{3} - 2 \beta_{2} - \beta_1 + 1) q^{59} + ( - 2 \beta_{3} - 6 \beta_{2} - \beta_1 - 3) q^{61} + ( - 2 \beta_{3} + \beta_{2} + \beta_1 + 6) q^{63} + (7 \beta_{3} - 2 \beta_{2} - 3 \beta_1 - 2) q^{65} + (3 \beta_{3} + 3 \beta_{2} - 1) q^{67} + ( - 6 \beta_{3} - 2 \beta_{2} - 2 \beta_1 - 6) q^{69} + ( - 3 \beta_{3} + 2 \beta_{2} + 3 \beta_1 + 1) q^{71} + (2 \beta_{3} - \beta_{2} - 4) q^{73} + ( - 3 \beta_{3} + 2 \beta_{2} + \beta_1 - 1) q^{75} + ( - 2 \beta_{3} - 2 \beta_{2} - 2 \beta_1) q^{77} + ( - 3 \beta_{3} - 3 \beta_{2} + 2 \beta_1 - 7) q^{79} + (4 \beta_{2} - 2 \beta_1 + 1) q^{81} + (3 \beta_{3} + 8 \beta_{2} - 3 \beta_1 + 7) q^{83} + (\beta_{3} + \beta_{2} - 4 \beta_1 - 4) q^{85} + ( - 2 \beta_{3} - \beta_{2} + 3 \beta_1 + 6) q^{87} + ( - 4 \beta_{3} - 4 \beta_{2} - \beta_1 - 6) q^{89} + ( - \beta_{3} - 8 \beta_{2} + 5 \beta_1 - 3) q^{91} + (6 \beta_{3} + 8 \beta_{2} + 6) q^{93} + (\beta_{3} - 2 \beta_{2} + 6 \beta_1 + 6) q^{97} + ( - 6 \beta_{3} - 5 \beta_{2} - 5 \beta_1 - 8) q^{99}+O(q^{100})$$ q + (-b3 - b2 - 1) * q^3 + (b3 - b2 - b1 - 1) * q^5 + (-b2 + b1) * q^7 + (2*b1 + 1) * q^9 + (-3*b2 - b1 - 2) * q^11 + (-b3 - b2 - b1 + 4) * q^13 + (b3 + 4*b2 + b1 + 1) * q^15 + (-b2 + 2*b1 + 1) * q^17 + (-2*b3 - 2*b2) * q^21 + (3*b3 - 3*b2 + 2*b1 + 1) * q^23 + (2*b3 - 3*b2) * q^25 + (-2*b2 - 2*b1) * q^27 + (-b3 - 3*b2 + b1 - 2) * q^29 + (b3 - b2 - 7) * q^31 + (4*b2 + 4*b1 + 6) * q^33 + (b3 + b2 - 2*b1 - 1) * q^35 + (-b3 - 2*b1 + 1) * q^37 + (-4*b3 - 3*b2 + 3*b1) * q^39 + (-2*b3 + 2*b2 - b1 - 2) * q^41 + (b3 + b2 - 2*b1 + 3) * q^43 + (-b3 + b2 - 3*b1 - 5) * q^45 + (-b3 + 4*b2 - b1 + 5) * q^47 + (-2*b3 - 3) * q^49 + (-4*b3 - 5*b2 - b1 - 2) * q^51 + (3*b3 - 2*b2 + 1) * q^53 + (5*b3 + b2 + 7) * q^55 + (-b3 - 2*b2 - b1 + 1) * q^59 + (-2*b3 - 6*b2 - b1 - 3) * q^61 + (-2*b3 + b2 + b1 + 6) * q^63 + (7*b3 - 2*b2 - 3*b1 - 2) * q^65 + (3*b3 + 3*b2 - 1) * q^67 + (-6*b3 - 2*b2 - 2*b1 - 6) * q^69 + (-3*b3 + 2*b2 + 3*b1 + 1) * q^71 + (2*b3 - b2 - 4) * q^73 + (-3*b3 + 2*b2 + b1 - 1) * q^75 + (-2*b3 - 2*b2 - 2*b1) * q^77 + (-3*b3 - 3*b2 + 2*b1 - 7) * q^79 + (4*b2 - 2*b1 + 1) * q^81 + (3*b3 + 8*b2 - 3*b1 + 7) * q^83 + (b3 + b2 - 4*b1 - 4) * q^85 + (-2*b3 - b2 + 3*b1 + 6) * q^87 + (-4*b3 - 4*b2 - b1 - 6) * q^89 + (-b3 - 8*b2 + 5*b1 - 3) * q^91 + (6*b3 + 8*b2 + 6) * q^93 + (b3 - 2*b2 + 6*b1 + 6) * q^97 + (-6*b3 - 5*b2 - 5*b1 - 8) * q^99 $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 4 q^{9}+O(q^{10})$$ 4 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 4 * q^9 $$4 q - 2 q^{3} - 2 q^{5} + 2 q^{7} + 4 q^{9} - 2 q^{11} + 18 q^{13} - 4 q^{15} + 6 q^{17} + 4 q^{21} + 10 q^{23} + 6 q^{25} + 4 q^{27} - 2 q^{29} - 26 q^{31} + 16 q^{33} - 6 q^{35} + 4 q^{37} + 6 q^{39} - 12 q^{41} + 10 q^{43} - 22 q^{45} + 12 q^{47} - 12 q^{49} + 2 q^{51} + 8 q^{53} + 26 q^{55} + 8 q^{59} + 22 q^{63} - 4 q^{65} - 10 q^{67} - 20 q^{69} - 14 q^{73} - 8 q^{75} + 4 q^{77} - 22 q^{79} - 4 q^{81} + 12 q^{83} - 18 q^{85} + 26 q^{87} - 16 q^{89} + 4 q^{91} + 8 q^{93} + 28 q^{97} - 22 q^{99}+O(q^{100})$$ 4 * q - 2 * q^3 - 2 * q^5 + 2 * q^7 + 4 * q^9 - 2 * q^11 + 18 * q^13 - 4 * q^15 + 6 * q^17 + 4 * q^21 + 10 * q^23 + 6 * q^25 + 4 * q^27 - 2 * q^29 - 26 * q^31 + 16 * q^33 - 6 * q^35 + 4 * q^37 + 6 * q^39 - 12 * q^41 + 10 * q^43 - 22 * q^45 + 12 * q^47 - 12 * q^49 + 2 * q^51 + 8 * q^53 + 26 * q^55 + 8 * q^59 + 22 * q^63 - 4 * q^65 - 10 * q^67 - 20 * q^69 - 14 * q^73 - 8 * q^75 + 4 * q^77 - 22 * q^79 - 4 * q^81 + 12 * q^83 - 18 * q^85 + 26 * q^87 - 16 * q^89 + 4 * q^91 + 8 * q^93 + 28 * q^97 - 22 * q^99

Basis of coefficient ring in terms of $$\nu = \zeta_{20} + \zeta_{20}^{-1}$$:

 $$\beta_{1}$$ $$=$$ $$\nu$$ v $$\beta_{2}$$ $$=$$ $$\nu^{2} - 3$$ v^2 - 3 $$\beta_{3}$$ $$=$$ $$\nu^{3} - 3\nu$$ v^3 - 3*v
 $$\nu$$ $$=$$ $$\beta_1$$ b1 $$\nu^{2}$$ $$=$$ $$\beta_{2} + 3$$ b2 + 3 $$\nu^{3}$$ $$=$$ $$\beta_{3} + 3\beta_1$$ b3 + 3*b1

## Embeddings

For each embedding $$\iota_m$$ of the coefficient field, the values $$\iota_m(a_n)$$ are shown below.

For more information on an embedded modular form you can click on its label.

Label $$\iota_m(\nu)$$ $$a_{2}$$ $$a_{3}$$ $$a_{4}$$ $$a_{5}$$ $$a_{6}$$ $$a_{7}$$ $$a_{8}$$ $$a_{9}$$ $$a_{10}$$
1.1
 1.90211 −1.17557 −1.90211 1.17557
0 −2.79360 0 −2.34458 0 1.28408 0 4.80423 0
1.2 0 −1.28408 0 3.69572 0 0.442463 0 −1.35114 0
1.3 0 −0.442463 0 −0.891491 0 −2.52015 0 −2.80423 0
1.4 0 2.52015 0 −2.45965 0 2.79360 0 3.35114 0
 $$n$$: e.g. 2-40 or 990-1000 Significant digits: Format: Complex embeddings Normalized embeddings Satake parameters Satake angles

## Atkin-Lehner signs

$$p$$ Sign
$$2$$ $$-1$$
$$19$$ $$1$$

## Inner twists

This newform does not admit any (nontrivial) inner twists.

## Twists

By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 5776.2.a.bt 4
4.b odd 2 1 722.2.a.n yes 4
12.b even 2 1 6498.2.a.bx 4
19.b odd 2 1 5776.2.a.bv 4
76.d even 2 1 722.2.a.m 4
76.f even 6 2 722.2.c.n 8
76.g odd 6 2 722.2.c.m 8
76.k even 18 6 722.2.e.s 24
76.l odd 18 6 722.2.e.r 24
228.b odd 2 1 6498.2.a.ca 4

By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
722.2.a.m 4 76.d even 2 1
722.2.a.n yes 4 4.b odd 2 1
722.2.c.m 8 76.g odd 6 2
722.2.c.n 8 76.f even 6 2
722.2.e.r 24 76.l odd 18 6
722.2.e.s 24 76.k even 18 6
5776.2.a.bt 4 1.a even 1 1 trivial
5776.2.a.bv 4 19.b odd 2 1
6498.2.a.bx 4 12.b even 2 1
6498.2.a.ca 4 228.b odd 2 1

## Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on $$S_{2}^{\mathrm{new}}(\Gamma_0(5776))$$:

 $$T_{3}^{4} + 2T_{3}^{3} - 6T_{3}^{2} - 12T_{3} - 4$$ T3^4 + 2*T3^3 - 6*T3^2 - 12*T3 - 4 $$T_{5}^{4} + 2T_{5}^{3} - 11T_{5}^{2} - 32T_{5} - 19$$ T5^4 + 2*T5^3 - 11*T5^2 - 32*T5 - 19 $$T_{7}^{4} - 2T_{7}^{3} - 6T_{7}^{2} + 12T_{7} - 4$$ T7^4 - 2*T7^3 - 6*T7^2 + 12*T7 - 4 $$T_{11}^{4} + 2T_{11}^{3} - 26T_{11}^{2} - 12T_{11} + 76$$ T11^4 + 2*T11^3 - 26*T11^2 - 12*T11 + 76 $$T_{13}^{4} - 18T_{13}^{3} + 109T_{13}^{2} - 232T_{13} + 61$$ T13^4 - 18*T13^3 + 109*T13^2 - 232*T13 + 61

## Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ $$T^{4}$$
$3$ $$T^{4} + 2 T^{3} - 6 T^{2} - 12 T - 4$$
$5$ $$T^{4} + 2 T^{3} - 11 T^{2} - 32 T - 19$$
$7$ $$T^{4} - 2 T^{3} - 6 T^{2} + 12 T - 4$$
$11$ $$T^{4} + 2 T^{3} - 26 T^{2} - 12 T + 76$$
$13$ $$T^{4} - 18 T^{3} + 109 T^{2} + \cdots + 61$$
$17$ $$T^{4} - 6 T^{3} - 9 T^{2} + 74 T - 19$$
$19$ $$T^{4}$$
$23$ $$T^{4} - 10 T^{3} - 50 T^{2} + \cdots - 1220$$
$29$ $$T^{4} + 2 T^{3} - 31 T^{2} - 92 T - 19$$
$31$ $$T^{4} + 26 T^{3} + 246 T^{2} + \cdots + 1436$$
$37$ $$T^{4} - 4 T^{3} - 19 T^{2} + 46 T - 19$$
$41$ $$T^{4} + 12 T^{3} + 19 T^{2} + \cdots - 359$$
$43$ $$T^{4} - 10 T^{3} + 10 T^{2} + \cdots - 100$$
$47$ $$T^{4} - 12 T^{3} + 4 T^{2} + 112 T + 76$$
$53$ $$T^{4} - 8 T^{3} - 31 T^{2} + 98 T + 181$$
$59$ $$T^{4} - 8 T^{3} + 4 T^{2} + 88 T - 164$$
$61$ $$T^{4} - 115 T^{2} + 150 T + 1025$$
$67$ $$T^{4} + 10 T^{3} - 30 T^{2} - 140 T - 20$$
$71$ $$T^{4} - 100 T^{2} + 360 T - 20$$
$73$ $$T^{4} + 14 T^{3} + 51 T^{2} + \cdots - 139$$
$79$ $$T^{4} + 22 T^{3} + 94 T^{2} + \cdots - 4724$$
$83$ $$T^{4} - 12 T^{3} - 196 T^{2} + \cdots - 6884$$
$89$ $$T^{4} + 16 T^{3} - 29 T^{2} + \cdots - 1159$$
$97$ $$T^{4} - 28 T^{3} + 99 T^{2} + \cdots - 7979$$